Cellular Quantitative StructureâActivity Relationship (Cell-QSAR

Article pubs.acs.org/jmc

Cellular Quantitative Structure−Activity Relationship (Cell-QSAR): Conceptual Dissection of Receptor Binding and Intracellular Disposition in Antifilarial Activities of Selwood Antimycins Senthil Natesan,†,‡ Tiansheng Wang,‡ Viera Lukacova,‡,§ Vladimir Bartus,‡ Akash Khandelwal,‡ Rajesh Subramaniam,† and Stefan Balaz*,† †

Albany College of Pharmacy and Health Sciences, Vermont Campus, Colchester, Vermont 05446, United States College of Pharmacy, Nursing and Allied Sciences, North Dakota State University, Fargo, North Dakota 58108, United States

‡

S Supporting Information *

ABSTRACT: We present the cellular quantitative structure−activity relationship (cell-QSAR) concept that adapts ligand-based and receptor-based 3D-QSAR methods for use with cell-level activities. The unknown intracellular drug disposition is accounted for by the disposition function (DF), a model-based, nonlinear function of a drug’s lipophilicity, acidity, and other properties. We conceptually combined the DF with our multispecies, multimode version of the frequently used ligand-based comparative molecular field analysis (CoMFA) method, forming a single correlation function for fitting the cell-level activities. The resulting cell-QSAR model was applied to the Selwood data on filaricidal activities of antimycin analogues. Their molecules are flexible, ionize under physiologic conditions, form different intramolecular H-bonds for neutral and ionized species, and cross several membranes to reach unknown receptors. The calibrated cell-QSAR model is significantly more predictive than other models lacking the disposition part and provides valuable structure optimization clues by factorizing the cell-level activity of each compound into the contributions of the receptor binding and disposition.

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INTRODUCTION Conceptual methods for the correlation of binding affinities with drug structure (3D-QSARs) are vital to the drug development process. The approaches include receptor-based methods,1 e.g., free energy perturbation,2 the linear response method,3−5 and the mining minima approach,6−9 and ligandbased methods for unknown receptors, e.g., comparative molecular field analysis (CoMFA).10 The methods have been developed and perform satisfactorily for the binding data measured with isolated macromolecules. Unfortunately, binding affinities for isolated macromolecules are often unavailable because the receptors have not yet been identified, cannot be isolated without denaturation or dissociation, or do not work in aqueous solutions. In such situations, more complex assays are used, deploying receptors reconstituted in lipid vesicles as the simplest system or utilizing intact cells, tissues, organs, or organisms. © 2012 American Chemical Society

The effective concentrations of the drugs in the receptor surroundings vary among the studied compounds because of interactions with nonreceptor assay constituents. Drug disposition has often been neglected in modeling the bioactivities measured in complex systems. At best, the 1octanol/water partition coefficients and their squares,11 and sometimes the dissociation constants,12 as properties affecting the disposition, have been included in simplified linear forms. In many cell-level studies, 3D-QSAR methods have been applied without any correction for ligand disposition or empirical models with various descriptors have been deployed. Here, we propose a straightforward solution to the problem of QSAR modeling of cell-level data: extend the proven 3D-QSAR methods by accounting for the varying ligand disposition in the receptor surroundings using the disposition function (DF). The Received: October 11, 2011 Published: April 2, 2012 3699

dx.doi.org/10.1021/jm201371y | J. Med. Chem. 2012, 55, 3699−3712

Journal of Medicinal Chemistry

Article

Table 1. Structures and Properties of Antimycin Analogues

compd no.

R1

R2

log Pa

pKab

log Kc

log DFc

log(1/EC50)d

1 2e 3 4 5 6 7 8 9 10 11e 12 13 14 15 16 17e 18 19f 20e 21 22 23 24e 25 26 27 28 29 30e 31f

3-NHCHO 3-NHCHO 5-NO2 5-SCH3 5-SOCH3 3-NO2 5-CN 5-NO2 3-SCH3 5-SO2CH3 5-NO2 5-NO2 5-NO2 5-NO2 3-SO2CH3 5-NO2 3-NHCHO 3-NHCHO 3-NHCOCH3 5-NO2 3-NO2 3-NO2-5-Cl 5-NO2 5-NO2 3-NO2 5-NO2 5-NO2 5-NO2 5-NO2 5-NO2 5-NO2

NHC14H29 NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H4-4-(OC6H4-4-CF3) NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H4-4-(OC6H5) NHC6H3-3-Cl-4-(COC6H4-4-Cl) NHC6H4-4-(OC6H3-2-Cl-4-NO2) NHC6H3-3-Cl-4-(OC6H4-4-OCH3) NHC6H3-3-Cl-4-(OC6H4-4-Cl) NHC6H3-3-Cl-4-(SC6H4-4-Cl) NHC6H13 NHC8H17 NHC14H29 NHC14H29 NHC14H29 NHC14H29 NHC6H4-4-C(CH3)3 NHC12H25 NHC16H33 NHC6H3-3-Cl-4-(NH-C6H4-4-Cl) NHC6H4-4-(OC6H4-3-CF3) NHC6H3-3-Cl-4-(NHC6H4-4-SCF3) NH-3-Cl-4-(3-CF3C6H4O)C6H3 NHC6H4-4-(CH(OH)C6H5) C6H4-4-Cl

7.491 5.955 6.944 7.402 5.722 6.944 6.698 6.541 7.402 5.731 5.658 6.264 5.884 6.150 5.731 7.476 3.259 4.317 7.468 8.648 8.648 9.419 5.386 7.590 9.706 6.900 6.541 7.899 7.114 3.870 4.329

7.88 7.21 5.15 8.00 6.80 4.60 5.66 5.15 7.47 6.11 5.31 5.37 5.20 5.11 5.10 5.25 7.88 7.88 7.88 5.83 5.28 4.68 5.55 5.83 5.28 5.12 5.16 5.04 5.06 5.42 6.73

1.181 2.307 1.710 2.290 1.843 1.668 2.534 1.483 1.931 1.681 1.450 1.803 1.936 1.839 1.433 1.554 4.018 4.091 1.181 1.016 1.293 1.684 1.659 1.647 0.555 0.968 1.675 1.711 1.872 1.197 0.777

4.150 3.438 5.476 4.086 3.535 5.548 5.225 5.397 4.292 4.166 4.818 5.184 5.067 5.271 5.034 5.504 0.430 1.488 4.143 5.406 5.522 5.564 4.369 5.361 5.524 5.477 5.394 5.541 5.510 3.015 2.221

5.155 5.620 7.398 6.319 5.125 6.824 7.839 7.022 6.420 6.000 6.097 7.131 6.921 6.770 6.301 7.357 6 have pKa > 6.5 (Table 1). However, lipophilic moieties including alkyl chains and two or three ring systems also affect the binding as well as disposition, so the structure−activity relationships are complex. Binding of Alkyl-Chain Analogues. The strongest binders, compounds 17 and 18 (Table 1, log K > 4) with a 3formylamino group in the salicylamide ring predominantly bind as neutral species in the closed conformation. The shorter alkyl chains may allow sufficient conformational freedom for these two compounds to make the best possible interactions in the binding site, leading to high binding affinities. Long-chain analogues (compounds 19−22 and 24, Table 1) predominantly bind in ionized open conformations or sometimes in ionized closed conformations, as seen for compounds 1 and 25. All long-chain analogues exhibit low to average binding affinity, possibly because of steric clashes of long chains in packed conformations (Figure 5d) with the sterically unfavorable regions in the midsection of the binding site. Binding of Two- and Three-Ring Analogues. Compounds with a 2-NO2 or 5-NO2 group in the salicylamide ring bind predominantly as ionized species, either in open or closed conformations, depending upon the presence of a 3-Cl substituent in the second ring (the absence of which seems to be compensated by the presence of a 2-Cl or 4-CF3 substituent in the third ring). Compounds 3, 6, 12, 14, 26, 28, and 29 with a 3-Cl substituent in the second ring mainly bind in the open conformation, although compounds 12 and 29 bind about equally in the closed conformation as well, and compound 28 shows slightly higher preference for the closed conformation. This variation is possibly affected by the presence of a carbonyl linker group between the second and third rings in compound 12 and 4-SCF3 and 3-CF3 groups in the third ring of compounds 28 and 29, respectively. Compounds without a 3-Cl substituent in the second ring (8, 11, 13, 23, 27, and 30) bind predominantly as ionized species and show higher preference for the closed conformation. The absence of steric hindrance by a chlorine substituent in the second ring allows these compounds to assume the preferred closed conformation with an intramolecular H-bond between the amide NH and phenoxide ion. Compound 16 (with a 5NO2 group in the first ring and a 3-Cl substituent in the second ring) prefers to bind in the closed conformation, possibly because of the presence of a thioether linkage between the second and third rings, resulting in a longer linker. The most active in this series, compound 7 with a 5-CN substituent in the salicylamide ring, binds predominantly as an ionized species in the open conformation. Influence of Disposition. The disposition of antimycins in female worms of Dipetalonema viteae after 120 h of exposure52 can be visualized for individual compounds as the difference between the experimental log(1/EC50) values and the log K values (Table 1) calculated from the affinity part of the DFMSMM CoMFA model (eq 1). These values are plotted as a function of the drugs’ lipophilicity (log P) and acidity (pKa) in Figure 7, along with the surface given by the disposition part of the DF-MSMM CoMFA model (eq 1) with optimized

Figure 7. Dependence of the pseudoequilibrium, membrane-based DF on lipophilicity and acidity, as given by the disposition part of eq 1 with the optimized coefficient values listed in the text. The data for best binders with poor disposition (compounds 17 and 18, Table 1) are shown as open points.

coefficients. The shape of log DF(P,Ka) is remarkably nonlinear: it is essentially composed of interconnected planes positioned at varying angles. No further curvatures are expected, according to the model, as the planes extend in all four directions to the limit property values, so this conceptual function safely predicts outside of the property ranges.33 Disposition significantly modulates the bioactivity of antimycins by ∼5 log(1/EC50) units. Using the correct form of the DF for the correlations is essential, because its peculiar shape (Figure 7) is difficult to emulate by polynomials, which are too flexible to approximate the planar areas.33 The resulting fit (eq 1, Table 3 and accompanying text) can be interpreted as follows. The variation in disposition is caused by pseudoequilibrium partitioning of ionizable compounds. The hypothetical receptors are localized in a membrane, with the drugs accessing the binding site (the CoMFA model in Figure 5) from within the bilayer. The optimized coefficients are associated with the processes that determine drug disposition: A, with bilayer accumulation and nonspecific protein binding; B, with accumulation in aqueous phases, with subscripts 1 and 2 indicating the relatedness to the nonionized and ionized species, respectively. The coefficient B2 was not necessary for a good fit, indicating that accumulation of ionized molecules in the aqueous compartments did not cause significant variation in overall disposition. The exponent, coming from eq 10 and containing coefficients C and D describing metabolism, need not be used in eq 1, meaning that C1 = C2 = D1 = D2 = 0. The model indicates that the pseudoequilibrium disposition of the studied compounds was not affected by metabolism, which could be either negligible or similar for all compounds. The internal pH value in terms f1 and f 2 (eq 12) was optimized as pH 7.56 ± 0.32. This high value can normally be encountered only in one part of the cell: in the matrix of mitochondria. Therefore, the fit indicates that the hypothetical receptors are localized in the bilayer that is in contact with the mitochondrial matrix. This indication agrees with the current knowledge about antimycin A’s mechanism of action, which focuses on inhibition of electron transport.57,66−69 Antimycin A binds at the ubiquinone binding site Qi of the complex that is located on the matrix side of the cristae membrane.70 Notably, the disposition function provides clear guidelines for the optimization of the drug structure for the best 3706



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part of the difficult-to-model Selwood data set that is often used as a test case for assessment of new variable-selection algorithms. The fitting results are very satisfactory and show that each component of the model (DF, MS, MM) improves the predictive ability. The optimized form of the disposition function indicates that the compounds bind to the hypothetical, membrane-bound receptor in the cristae bilayer of mitochondria and the differences in their metabolic rates are negligible. The perceived complexity of the Selwood data set may be explained by the nonlinearity of the model in lipophilicity and acidity in combination with the lack of proper descriptors for acidity in the original set, the peculiar intramolecular H-bonds in the salicylamide moiety that differ for neutral and ionized species, and the structure-specific receptor binding. The results also illustrate the utility of the cell-QSAR approach in medicinal chemistry. Specifically, the cell-QSAR approach delineates the affinity and disposition contributions to bioactivity, which provide for a straightforward, separate structure optimization for affinity and disposition. Although further testing is necessary, the cell-QSAR concept seems to be a promising direction in the structure-based or ligand-based predictions of bioactivities in cellular or more complex systems and drug structure optimization. For larger and more diverse data sets, better superposition techniques and methods for prediction of species fractions will be necessary. The model-based nature opens the cell-QSAR concept for incorporation of any future improvements in 3D-QSAR techniques and the disposition function, as well as new pharmacology and pharmacodynamic scenarios.

disposition. The top plateau in Figure 7 indicates the range of log P and pKa values that lead to the maximum drug disposition in the receptor surroundings. The region of the maximum disposition can be defined as pKa ≤ log P for log P ≤ 6 and pKa ≤ 6 for log P > 6. The knowledge of suitable combinations of log P and pKa values provides unique clues for optimization of drug structure, also because the lower log P and pKa values will lead to increased aqueous solubility. Factoring Bioactivity into Affinity and Disposition Components. Equation 1 defines two contributions to the affinity: log K (the first term) and disposition (log DF; the remaining terms). The two contributions were enumerated from the calibrated DF-MSMM CoMFA model (eq 1 and the accompanying text, Table 3), summarized in Table 1, and plotted against the experimental antifilarial activities (Table 1) in Figure 8.

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METHODS

Pharmacokinetics (PK), Pharmacology (PL), and Pharmacodynamics (PD) Phases of the Drug Effect in QSAR. The PK phase is described by the DF. The simplest PL/PD scenario consists of a situation when the PL phase is represented by the 1:1, fast, and reversible ligand−receptor interaction and the PD phase describes the biological effect, which is a direct and immediate consequence of the drug−receptor interaction and is proportional to the fraction of occupied receptors. The ligand−receptor association constant K is defined using the concentrations (denoted by square brackets) of the ligand (L), receptor (R), and complex (LR) as K = [LR]/([L][R]). In a cellular system, the free ligand concentration [L] varies because of the disposition, and consequently, all three concentrations are timedependent. Usually, the concentrations are sufficiently low to approximate activities. If the concentration of the free receptors [R] is expressed as the difference between the total receptor concentration [R]0 and [LR], the fraction of occupied receptors is given as [LR]/ [R]0 = K[L]/([L] + 1) = E/Emax. The latter equality comes from the simplest PD phase, describing the primary effect E as being proportional to the fraction of occupied receptors, [LR]/[R]0. Secondary effects lacking this simple tie to the receptor modification require more complex PD models.71 Nevertheless, the above scenario applies to many drug effects caused by competitive enzyme inhibition or receptor binding. Since the receptor-bound drug represents only a small drug fraction in the body, the disposition is not significantly affected by the receptor binding, and the ligand concentration in the receptor surroundings, [L], can be expressed using the DF with properties p and time t as variables and the initial ligand concentration c0:72

Figure 8. Dissection of affinity (log K; red symbols) and disposition (log DF; blue symbols) contributions to overall bioactivity by the calibrated DF-MSMM CoMFA model. Stars mark the best binders 17 and 18 with poor disposition. All data are given in Table 1.

Figure 8 and Table 1 show that two weakly active compounds (17 and 18) are actually the best binders, which are held back by poor disposition. Their disposition can be, in principle, increased by 4−5 logarithmic units, leading to overall bioactivities approaching log(1/EC50) ≈ 9.5, which is almost 2 orders of magnitude better than that of the most active compound 7. Disposition is easier to optimize than binding because the former depends on lipophilicity and acidity, which are well understood in terms of drug structure. The DFMSMM CoMFA model, and cell-QSAR models in general, provides unique clues for optimization of the compounds.

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CONCLUSIONS The cell-QSAR concept adapts ligand-based and receptor-based 3D-QSAR approaches for use with the bioactivities measured in cells or more complex systems. The adaptation consists of the extension of the correlation equation specific for the given 3DQSAR method by the DF. The DF describes changes in the concentration of compounds in the receptor surroundings originating from membrane accumulation, binding to nonreceptor proteins, hydrolysis, and possibly other processes. The cell-QSAR concept was used here with the popular ligandbased CoMFA method. A conceptual combination of the DF with our MSMM extension of the CoMFA model was applied to antifilarial activities of antimycin A analogues, representing a

[L] = DF(p , t )c0

(2)

To obtain a PK/PL/PD expression, [L] in the above expression for E/Emax is replaced by the product DF(p,t) cX(t), where cX(t) are the isoeffective drug concentrations eliciting the effect, which is equal to the fraction X of the maximum effect. With E/Emax = X, the bioactivity, characterized by cX(t), is described as72 3707



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(3)

The experimentally measured Ki reflects the total concentration of

Equivalent PK/PL/PD dependencies were derived for the irreversible drug−receptor interactions, either single or preceded by a fast, reversible step.72 Equation 3 and equivalent PK/PL/PD relationships provide the blueprint for a conceptual combination of the disposition function with any ligand-based or receptor-based 3D-QSAR method for prediction of binding affinities. The K term in eq 3 needs to be replaced by a suitable conceptual 3D-QSAR correlation equation, and the disposition function can be selected from the available repertoire.30,32 This cell-QSAR concept represents the recipe for the formulation of conceptual QSARs for cell-based assays and other complex systems. Multispecies Multimode Equilibria. Studied antimycin analogues ionize under the conditions of the experiments, and each species forms different intramolecular H-bonds. The molecules are flexible, and each species can bind to the receptor R in different orientations or conformations (modes). The generalized multispecies multimode binding equilibria are outlined in the following equation:

the ligand/receptor complexes in any species and any binding mode. A

1/cX = K[DF(p , t )]X /(1 − X )

series of rearrangements to the definition of Ki provides the desired

[LR i] [Li][R]

Ki =

s

=

∑ j=1

= ∑ fij j=1 s

=

[LR ij] [Lij][R] m

∑ fij ∑ j=1

k=1

[LR ijk] [Lij][R]

(6)

function of Kijk, shown in eq 6: (1) the total concentration of the complex [LRi] is expressed as the sum of bound concentrations of individual species, (2) each summand is formally multiplied by [Lij]/ [Lij] to introduce the fraction of each species, f ij = [Lij]/[Li], and (3) the bound concentration of each species [LRij] is broken down to individual binding modes. The relationship between the microscopic and total association constants is obtained by expressing the msummands in eq 6 using eq 5: s

m

∑ fij ∑ K ijk j=1

(7)

k=1

Here f ij is the fraction of the jth molecular species in the ith compound in solution. For compounds present in the solution as one species, the right-hand side of eq 7 is equal to the k-summation. In this case, eq 7 is in accordance with published analyses of formally analogous situations: the statistical thermodynamic73 and equilibrium58,59 treatment of multimode binding in ligand/protein interactions and kinetic analyses of a reversible unimolecular reaction leading to different products74 or isomers.75 The expressions for the fractions f ij of the jth species can be derived from the definition of the ionization constants. Only one molecule of a compound, in any species and any mode, can be bound in each complex. The distribution of species and modes in the bound state is characterized by their prevalences. For the kth mode of the jth bound species for the ith compound, the prevalence is

Each binding equilibrium is characterized by the microscopic association constant Kijk (eq 5) and represents the receptor binding of the ith ligand in the jth species and the kth binding mode. In principle, the number of species s and the number of binding modes m can differ for individual ligands. In the studied case, there are only two species (s = 2, nonionized and ionized, with charges 0 and −1, respectively) for each ligand (Table 1). The number of modes is unknown, since there is no experimental information available on either the receptor or the complex, and can differ for individual ligands. In the general expressions, s and m represent the maximum numbers of species and binding modes for the tested ligands. The missing species or modes of some ligands will have zero fractions f ij or zero association constants Kijk (see below).

[LR ijk] [LR i]

=

K ijk[Lij][R] K i[Li][R]

=

K ijkfij Ki

≈

K ijkfij s

m

Σ fij Σ K ijk

(8)

j=1 k=1

The numerator and denominator come from eqs 5 and 6, respectively. The third quasi-equality ensures that the sum of all s × m prevalences equals unity. The prevalence of the jth bound species in all modes and the prevalence of the kth bound mode in all species can easily be calculated using eq 8. For the application in CoMFA, eq 7 needs to be logarithmized and the log Kijk values must be replaced by the correlation equation for CoMFA: s

m

log K i = log ∑ fij

The microscopic association constant for the ith molecule’s jth species in the kth binding mode is

j=1

∑ 10 k=1

s log K ijk

= log ∑ fij j=1

m

2g

∑ ClXijkl + C0

∑ 10l=1 k=1

(9) Here Cl are the regression coefficients characterizing the weights of the ligand−probe interaction energies X in individual cross-sections of a grid surrounding the superimposed ligands. The subscripts indicate the relatedness to the ith compound, jth species, kth mode, and lth grid point, and g is the total number of grid points. Other contributions to the binding free energy (internal ligand energy, binding entropy, desolvation) can be added to the summation in the exponents. In this case, we only used the CoMFA energies because the size of the used

[LR ijk] [Lij][R]

[Li][R]

s

Ki =

K ijk =

[LR ij]

(5)

Usually, only the total association constant of the ith ligand (Ki) is determined, although the microscopic association constants Kijk are the relevant quantities to be correlated with the structure. To obtain a conceptual correlation equation for Ki, it must be expressed as a function of Kijk. 3708



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The interaction of a rare species with the receptor can release about 2−3 kcal/mol more free energy than that of the prevalent species because both species bind by weak interactions and differ in a single, well-defined structural feature. For this situation, the product f ijKij for the rare species would be less than 1% of the same quantity for the prevalent species. Superposition. Pharmacophore models were built using the Galahad procedure54,55 in Sybyl-X. The ligands are first aligned flexibly to each other in torsional space, and then the conformations produced are rigidly aligned in Cartesian space. Pharmacophore and steric bitmaps (TUPLET78) representing H-bonding and steric terms, respectively, along with the total ligand energy, serve as input to a multiobjective fitness function. Produced pharmacophores indicated that the salicylamide moieties of all compounds in all species are superimposed in all cases. In the absence of structural and other information on the receptor, the most active and, at the same time, one of most rigid analogues, compound 7 (Figure 2, Table 1), was selected as the template for superposition. Conformational analysis was performed using the Tripos force field in Sybyl79 with an increment of 10° for all considered torsions. When the molecule forms an intramolecular H-bond (Figure 1), either as a neutral species (OH···OC) or as an ionized species (NH···O−), the first torsion Φ1 is fixed and only Φ2, Φ3, and Φ4 are allowed to rotate in conformational analysis. In the opposite case, when analogues have no intramolecular H-bond, all four torsions in Figure 2 are considered rotatable. The approximate minimum conformations were further optimized and Mulliken charges were calculated in Jaguar80 using the DFT/B3LYP method with a 6-31G** basis set. Charge and spin multiplicity were entered manually for different protonation states. In this way, the conformations of the template for the superposition were obtained. Open conformation, the mode without the intramolecular H-bond has the same optimal geometries for both neutral and ionized species. In the MM approach, the user is allowed to enter several possible bound conformations of the template in one optimization run. The MSMM approach extends this option to all species which are present in the solution surrounding the receptor. The four basic template molecules of compound 7 were superimposed by the atom fit method in Sybyl (Figure 3a). All studied compounds do not share a common skeleton; therefore, for each compound, both species in all modes were superimposed on the templates using the FlexS procedure65 as incorporated in Sybyl (Figure 3c). FlexS combines conformational search with alignment in two steps. First, a base fragment of the ligand is selected and placed on the template. Second, as the rest of the ligand parts are incrementally built, torsion angles are perturbed to generate new conformations, and each conformation is scored for superposition quality. The obtained conformations of all compounds were minimized by Jaguar using the DFT/B3LYP method81 with a 631G** basis set82,83 with all torsions fixed. The Mulliken charges84 were calculated for the optimized geometries. The superposition was repeated, and the resulting conformations were used in the CoMFA analyses. Interaction Energies. Steric and electrostatic interaction energies Xijkl with an sp3 carbon probe with a +1 charge were calculated for each compound (subscript i) in each species (subscript j) and each binding mode (skeleton conformations × ring-flipping conformations, subscript k). The probe was placed at each lattice intersection (subscript l) of a 3D grid (2 Å in each coordinate direction). The grid has the following coordinates: −7 to +18 Å along the x-axis, 1−21 Å along the y-axis, and −15 to +11 Å along the z-axis. An energy cutoff of 30 kcal/ mol was used to cap the maximum used energies. Regression Analysis. The correlation equation is obtained by taking the logarithm of eq 3, in which the association constant log K is replaced by eq 9 and the disposition function DF(p,t) is given by eq 10, as combined with eq 11, in which the fractions are expressed by eq 12. The receptor-binding and disposition components have, from the optimization viewpoint, quite different properties: eq 9 has only one form, and the descriptive variables need to be selected from a large pool, while eqs 10−12 contain only two variables, which are organized in one of two possible functional forms, depending upon the location

data set did not allow a rigorous optimization of more than a dozen coefficients. Intracellular Disposition. The pseudoequilibrium DF expresses the time course of the concentration of nonionized ligand molecules in the aqueous phases of the studied biosystem during the elimination period:

DF(p , t ) =

β β 1 e−(CP + D)t /(AP + B) AP + B

β

(10)

Here P is the reference partition coefficient, β is the empirical Collander coefficient, and t is the exposure time. The terms A, B, C, and D characterize the key fate-determining processes the ligand molecules undergo in the biological system: A, accumulation in membranes/protein binding; B, distribution in aqueous phases; C, lipophilicity-dependent elimination; D, lipophilicity-independent elimination. For the receptors localized in the bilayer, eq 10 needs to be multiplied by Pβ. If compounds are present in the aqueous phase in the receptor surroundings in s species created by ionization, tautomerism, or carbonyl hydration, the model suggests that each of the parameters A, B, C, and D (represented by Y) from eq 10 be expanded as76 s

Y=

∑ f j Yj

(11)

j=1

The fractions f of individual species are the same quantities as in eq 9, except the subscripts i, indicating the compound, are omitted. The subscripts j denote, as in eq 9, the quantities associated with the jth species. The fractions of nonionized, neutral molecules (subscript j = 1) and ionized molecules with an overall charge of −1 (subscript j = 2) are given by the definition of the dissociation constant Ka for the given pH of the medium and can be written, respectively, as

f1 =

10 pKa − pH 1 + 10 pKa − pH

and

f2 =

1 1 + 10 pKa − pH

(12)

Generally, on the basis of the probable intracellular localization of the receptors, a reasonable intracellular pH range is 5−8, the lower limit being observed in lysozomes and the upper limit in the mitochondrial matrix. The pH value of the cultivation medium could affect these values, but the exact pH of the assay was not published.52 For this reason and because the receptor localization is unknown, the pH value in eq 12 was treated as one of the adjustable coefficients. Data Set. The Selwood data set includes in vitro filaricidal activities against D. viteae of 31 antimycin analogues (Table 1), which were determined by monitoring the leakage of incorporated radiolabeled adenine from intact female D. viteae after 120 h of exposure.36,52 The EC50 values for adenine leakage were calculated after 120 h of exposure to a range of drug concentrations at 37 °C under neutral conditions.52 The 1-octanol/water partition coefficients P were calculated by the ClogP program in Bio-Loom51 software for neutral molecules and are summarized in Table 1. The values differ from the original data77 because of the developments in the ClogP software. The differences are within ±0.25 log unit, except for compounds 6 (Table 1, log Pold − log Pnew = 1.526), 10 (−0.770), 12 (−0.638), 14 (−1.654), and 26 (0.325). The log P values were used with the belief that they properly reflect the relative lipophilicity of the studied compounds, although the values of log P > 7 were marked in the ClogP output as unrealistically high. Ionization decreases the log P values by several units, and most compounds were present mainly as ionized species under physiologic conditions. Our multispecies approach can take into account tautomers and hydrates of aldehydes and ketones; hence, formation of these species was examined using the SPARC Web server50 and was found negligible for all compounds on the basis of the following reasoning. As shown in eq 7, the binding contribution of each species is given by the product of the association constant Kij (represented by the sum of the association constants Kijk for individual modes61) and the species fraction f ij. The fraction limit for a species to be included in the correlation was set to 10−4 on the basis of the following considerations. 3709



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of the receptors either in the aqueous phase or in membranes. For the receptor-binding part (eq 9), the coefficients Cl are associated with the grid points and are independent of the species and modes. Therefore, the final set of optimized coefficients is independent of the numbering of individual species and modes. Notably, the inclusion of multiple species and multiple modes does not increase the number of optimized regression coefficients Cl. The correlation equation is nonlinear in the coefficients Cl in eq 9 and the coefficients A−D, pH and β in eqs 10−12. All statistical procedures61 were written in C programming language and integrated in Sybyl through the Sybyl Programming Language (SPL) scripts. The choice of one of the two implemented optimization methods depends on the total number (N) of coefficients C in the calibrated model and on the number of compounds (n). Nonlinear regression analysis is applied when N ≤ n/2. Otherwise, the developed software automatically switches to the PLS analysis with a linearized form61 of eq 9. The second option was not used for optimization of the best models. Both optimization methods are sensitive to good initial estimates. Therefore, a forward-selection procedure, conceptually similar to that used before for multimode analysis,61 was a rational choice. Before analysis, the Xijkl variables (energy columns) were sorted using the following criteria: (1) high and sustained variability, (2) the minimal number of colinear columns with similar information, and (3) even distribution of selected grid points around the ligands. Sustained variability means that the energy values cover the whole interval more or less evenly and are not clustered. An extreme case is the situation when all energies in a given column are equal (usually zero) and only one mode has a different energy. These singularities originate when only one molecule protrudes into a subspace and other molecules do not affect the probe interaction energies in that subspace. Singular molecules need to be eliminated from the analysis because they do not provide statistically significant information about the poorly covered subspace. The model calibration can be steered by user-defined inputs, for which the default values are shown in parentheses. The process starts with a quick scan through the models consisting of a few (four) variables which are randomly selected from the top variables (10%). The initial values of the coefficients (±0.1) are systematically evaluated by a grid search. In this step, no optimization is applied and the correlation equation is evaluated for the initial coefficients. For the best models (top 10% r2), the coefficients are optimized. At each of the following steps, a reduction of the number of coefficients is attempted: the coefficients, which do not lead to a decrease in the r2 value, are eliminated. The best models (top 5% r2) are developed by a gradual addition of groups of a few (three) variables until all variables from the selected set (top 10%) are used. The procedure is finalized by finetuning of the best models (top 5% r2) by addition of single variables. The best model is selected on the basis of the following criteria: N, r2, q2 for the leave-one-out cross-validation (applied to the training set only to eliminate unstable models), and the standard errors of the parameters.

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ACKNOWLEDGMENTS

This work was supported in part by NIH NIGMS Grant R01 GM80508, NIH NCRR Grants P20 RR 15566 and P20 RR 16471, the NSF EPSCoR EPS-0814442 program, and access to Teragrid computation resources for Projects MCB100078 and MCB110017.

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ABBREVIATIONS USED A, disposition function term for accumulation in lipids and protein binding; β, Collander coefficient; B, disposition function term for accumulation in aqueous phases; C and D, disposition function terms for elimination; c0, initial concentration; cX, isoeffective concentration; Cl, regression coefficients in the CoMFA expression; DF, disposition function; E, drug effect; Emax, maximum drug effect; f, species fraction; g, number of grid points; K, association constant; L, ligand; LR, ligand− receptor complex; m, number of modes; MM, multimode; MS, multispecies; n, number of compounds; N, number of coefficients; P, partition coefficient; PD, pharmacodynamics; PK, pharmacokinetics; PL, pharmacology; PRESS, predictive sum of squares of deviations; q2, squared predictive correlation coefficient; r2, squared descriptive correlation coefficient; R, receptor; s, number of species; SM, single mode; SS, single species; SSE, sum of squares of errors; SYY, sum of squares of deviations from the average; X, interaction energy

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REFERENCES

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ASSOCIATED CONTENT

* Supporting Information S

Prevalences of all species and modes as described in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: (802) 735-2615. Fax: (802) 654-0716. Present Address §

Simulations Plus, Inc., Lancaster, CA 93534.

Notes

The authors declare no competing financial interest. 3710



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